This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A174220 #12 Apr 03 2023 10:36:12 %S A174220 1,2,2,4,3,8,2,6,8,5,22,15,7,10,23,21,6,26,19,7,34,18,33,38,27,18,27, %T A174220 12,30,95,29,59,14,79,11,59,58,37,61,59,23,96,22,43,19,131,143,50,31, %U A174220 55,84,30,134,86,88,77,24,87,60,28,162,227,73,37,55,248,104,174,39,65,104,143 %N A174220 Number of regular primes between p^2 and p*q, where p and q are consecutive primes. %C A174220 If you graph a(n) versus n, an interesting pattern with random-looking fluctuations emerges. %C A174220 As you go farther along the n-axis, greater are the number of regular primes, on average, within each interval obtained. %C A174220 The smallest count of 1 occurs only once at the very beginning. %C A174220 I suspect all numbers in this sequence are > 0. %C A174220 If one could prove that there is at least 1 regular prime within each interval, this would imply that regular primes are infinite. %C A174220 This would be very significant since "Kummer was able to prove Fermat's Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles's recent work was the only demonstration of Fermat's Last Theorem for a large class of exponents." (see Jao link). %H A174220 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/Regular.html">Regular prime</a> %H A174220 D. Jao, <a href="https://planetmath.org/regularprime">Regular prime</a>, PlanetMath.org. %F A174220 Used the table of irregular primes by T.D. Noe in A000928 to extract a longer list of regular primes from a list of odd primes. %e A174220 Take any pair of consecutive primes. Say the first (2,3). Square the first term, and then take the product of the two to obtain an interval (4,6). Within this interval, there is 1 regular prime, which is 5. Hence the very first term of the sequence above is 1. Similarly, the second term, 2, refers to the two regular primes 11 and 13. %o A174220 (PARI) has(p)=forstep(k=2, p-3, 2, if(numerator(bernfrac(k))%p==0, return(0))); 1 %o A174220 a(n,p=prime(n))=my(q=nextprime(p+1),s); forprime(r=p^2,p*q, if(has(r), s++)); s \\ _Charles R Greathouse IV_, Nov 02 2022 %Y A174220 Cf. A007703, A000928. %K A174220 nonn %O A174220 1,2 %A A174220 _Jaspal Singh Cheema_, Mar 12 2010 %E A174220 New name from _Charles R Greathouse IV_, Nov 02 2022